A129803 -id:A129803 - cp's OEIS frontend

A034706 Numbers which are sums of consecutive triangular numbers.

0, 1, 3, 4, 6, 9, 10, 15, 16, 19, 20, 21, 25, 28, 31, 34, 35, 36, 45, 46, 49, 52, 55, 56, 64, 66, 74, 78, 80, 81, 83, 84, 85, 91, 100, 105, 109, 110, 116, 119, 120, 121, 130, 136, 144, 145, 153, 155, 161, 164, 165, 166, 169, 171, 185, 190, 196, 199, 200, 202, 210

Offset: 1

Erich Friedman

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
D. Subramaniam, E. Trevino, and P. Pollack, On sums of consecutive triangular numbers, INTEGERS 20A (2020) A15.

Complement gives A050941.

Cf. A000217 (1 consec), A001110 (2 consec), A129803 (3 consec), A131557 (5 consec), A257711 (7 consec), A034705, A269414 (subsequence of primes).

-- import Data.Set (deleteFindMin, union, fromList); import Data.List (inits)
a034706 n = a034706_list !! (n-1)
a034706_list = f 0 (tail $ inits $ a000217_list) (fromList [0]) where
   f x vss'@(vs:vss) s
     | y < x = y : f x vss' s'
     | otherwise = f w vss (union s $ fromList $ scanl1 (+) ws)
     where ws@(w:_) = reverse vs
           (y, s') = deleteFindMin s
-- Reinhard Zumkeller, May 12 2015

isA034706 := proc(n)
    local a,b;
    for a from 0 do
        if a*(a+1)/2 > n then
            return false;
        end if;
        for b from a do
            tab := (1+b-a)*(a^2+b*a+a+b^2+2*b)/6 ;
            if tab = n then
                return true;
            elif tab > n then
                break;
            end if;
        end do:
    end do:
end proc:
for n from 0 to 100 do
    if isA034706(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 14 2015

M = 1000; (* to get all terms <= M *)
nmax = (Sqrt[8 M + 1] - 1)/2 // Ceiling;
Table[Sum[n(n+1)/2, {n, j, k}], {j, 0, nmax}, {k, j, nmax}] // Flatten // Union // Select[#, # <= M&]& (* Jean-François Alcover, Mar 10 2019 *)

A131557 Triangular numbers that are the sums of five consecutive triangular numbers.

Original entry on oeis.org

55, 2485, 17020, 799480, 5479705, 257429395, 1764447310, 82891465030, 568146553435, 26690794309585, 182941425758080, 8594352876220660, 58906570947547645, 2767354935348742255, 18967732903684582930, 891079694829418784770, 6107551088415488155135

Offset: 1

Richard Choulet, Oct 06 2007

			a(1) = 55 = 3+6+10+15+21.

Alois P. Heinz, Table of n, a(n) for n = 1..250
Index entries for linear recurrences with constant coefficients, signature (1,322,-322,-1,1).

Cf. A129803.

a:= n-> `if`(n<2, [0, 55][n+1], (<<0|1|0>, <0|0|1>, <1|-323|323>>^iquo(n-2, 2, 'r'). `if`(r=0, <<2485, 799480, 257429395>>, <<17020, 5479705, 1764447310>>))[1, 1]): seq(a(n), n=1..20); # Alois P. Heinz, Sep 25 2008, revised Dec 15 2011

LinearRecurrence[{1, 322, -322, -1, 1}, {55, 2485, 17020, 799480, 5479705}, 20] (* Jean-François Alcover, Oct 05 2019 *)

The subsequences with odd indices and even indices satisfy the same recurrence relations: a(n+2) = 322*a(n+1) - a(n) - 680 and a(n+1) = 161*a(n) - 340 + 9*sqrt(320*a(n)^2 - 1360*a(n) - 175).
G.f.: -5*x*(11+486*x-635*x^2+2*x^4) / ( (x-1)*(x^2+18*x+1)*(x^2-18*x+1) ).
8*a(n) = 17 + 45*A007805(n) + 18*(-1)^n*A049629(n). - R. J. Mathar, Apr 28 2020

More terms from Alois P. Heinz, Sep 25 2008
a(6) and a(8) corrected by Harvey P. Dale, Oct 02 2011
a(10), a(12), a(14) corrected at the suggestion of Harvey P. Dale by D. S. McNeil, Oct 02 2011

A257711 Triangular numbers (A000217) that are the sum of seven consecutive triangular numbers.

Original entry on oeis.org

210, 3486, 51681, 883785, 13125126, 224476266, 3333728685, 57016086141, 846753959226, 14481861401910, 215072171913081, 3678335779997361, 54627484911961710, 934282806257926146, 13875166095466359621, 237304154453733242085, 3524237560763543380386

Offset: 1

Colin Barker, May 05 2015

			210 is in the sequence because T(20) = 210 = 10+15+21+28+36+45+55 = T(4)+ ... +T(10).

Colin Barker, Table of n, a(n) for n = 1..830
Index entries for linear recurrences with constant coefficients, signature (1,254,-254,-1,1).

Cf. A000217, A001110, A129803, A131557, A257712, A257713, A259413, A259414, A259415.

LinearRecurrence[{1, 254, -254, -1, 1}, {210, 3486, 51681, 883785, 13125126}, 30] (* Vincenzo Librandi, Jun 27 2015 *)

Vec(-21*x*(x^4-245*x^2+156*x+10) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)) + O(x^100))

G.f.: -21*x*(x^4-245*x^2+156*x+10) / ((x-1)*(x^2-16*x+1)*(x^2+16*x+1)).

16*a(n) = 104 +225*A157456(n+1) +7*(-1)^n*A159678(n+1). - R. J. Mathar, Apr 28 2020

A257712 Triangular numbers (A000217) that are the sum of eight consecutive triangular numbers.

Original entry on oeis.org

120, 276, 1176, 28920, 126756, 306936, 1345620, 33362196, 146264856, 354192420, 1552832856, 38499933816, 168789505620, 408737734296, 1791967758756, 44428890250020, 194782943209176, 471682991173716, 2067929240760120, 51270900848577816, 224779347673872036

Offset: 1

Colin Barker, May 05 2015

			120 is in the sequence because T(15) = 120 = 1+3+6+10+15+21+28+36 = T(1)+ ... +T(8).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1154,-1154,0,0,-1,1).

Cf. A000217, A001110, A129803, A131557, A257711, A257713, A259413, A259414, A259415.

LinearRecurrence[{1, 0, 0, 1154, -1154, 0, 0, -1, 1}, {120, 276, 1176, 28920, 126756, 306936, 1345620, 33362196, 146264856}, 30] (* Vincenzo Librandi, Jun 27 2015 *)
Select[Total/@Partition[Accumulate[Range[5*10^6]],8,1],OddQ[ Sqrt[ 1+8#]]&] (* The program generates the first 16 terms of the sequence *) (* Harvey P. Dale, Feb 27 2022 *)

Vec(-12*x*(3*x^8+7*x^6+13*x^5-3387*x^4+2312*x^3+75*x^2+13*x+10) / ((x-1)*(x^2-6*x+1)*(x^2+6*x+1)*(x^4+34*x^2+1)) + O(x^100))

G.f.: -12*x*(3*x^8+7*x^6+13*x^5-3387*x^4+2312*x^3+75*x^2+13*x+10) / ((x-1)*(x^2-6*x+1)*(x^2+6*x+1)*(x^4+34*x^2+1)).

A257713 Triangular numbers (A000217) that are the sum of ten consecutive triangular numbers.

Original entry on oeis.org

1485, 7260, 28920, 142845, 2112540, 10440165, 41673885, 205953660, 3046252485, 15054681960, 60093684540, 296985006165, 4392693942120, 21708840917445, 86655051404085, 428252172907560, 6334261618255845, 31304133548245020, 124956524030977320, 617539336347666645

Offset: 1

Colin Barker, May 05 2015

nonn

			1485 is in the sequence because T(54) = 1485 = 78+91+105+120+136+153+171+190+210+231 = T(12)+ ... +T(21).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1442,-1442,0,0,-1,1).

Cf. A000217, A001110, A129803, A131557, A257711, A257712, A259413, A259414, A259415.

 LinearRecurrence[{1, 0, 0, 1442, -1442, 0, 0, -1, 1}, {1485, 7260, 28920, 142845, 2112540, 10440165, 41673885, 205953660, 3046252485}, 30] (* Vincenzo Librandi, Jun 27 2015 *)

Vec(-15*x*(8*x^8-5*x^7+5*x^5-11445*x^4+7595*x^3+1444*x^2+385*x+99) / ((x-1)*(x^2-6*x-1)*(x^2+6*x-1)*(x^4+38*x^2+1)) + O(x^100))

G.f.: -15*x*(8*x^8-5*x^7+5*x^5-11445*x^4+7595*x^3+1444*x^2+385*x+99) / ((x-1)*(x^2-6*x-1)*(x^2+6*x-1)*(x^4+38*x^2+1)).

A259413 Triangular numbers (A000217) that are the sum of eleven consecutive triangular numbers.

Original entry on oeis.org

2145, 3916, 7381, 13530, 843051, 1547920, 2926990, 5374281, 335521560, 616057651, 1164924046, 2138939715, 133536727236, 245189386585, 463636832725, 851292621696, 53147281907775, 97584759792586, 184526294489911, 338812324484700, 21152484662556621

Offset: 1

Colin Barker, Jun 26 2015

			2145 is in the sequence because T(65) = 2145 = 105 + 120 + 136 + 153 + 171 + 190 + 210 + 231 + 253 + 276 + 300 = T(14) + ... + T(24).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,398,-398,0,0,-1,1).

Cf. A000217, A001110, A129803, A131557, A257711, A257712, A257713, A259414, A259415.

LinearRecurrence[{1, 0, 0, 398, -398, 0, 0, -1, 1}, {2145, 3916, 7381, 13530, 843051, 1547920, 2926990, 5374281, 335521560}, 30] (* Vincenzo Librandi, Jun 27 2015 *)

Vec(-11*x*(6*x^8-x^7+x^5-2199*x^4+559*x^3+315*x^2+161*x+195)/((x-1)*(x^4-20*x^2+1)*(x^4+20*x^2+1)) + O(x^30))

G.f.: -11*x*(6*x^8-x^7+x^5-2199*x^4+559*x^3+315*x^2+161*x+195) / ((x-1)*(x^4-20*x^2+1)*(x^4+20*x^2+1)).

A259414 Triangular numbers (A000217) that are the sum of thirteen consecutive triangular numbers.

Original entry on oeis.org

2080, 414505, 28815436, 49317346, 3428789455, 698283666730, 48548229019381, 83089887991201, 5776831256176630, 1176469718198438755, 81794153348207147926, 139990009467226925656, 9732816854065394603605, 1982118534159467652450580, 137806953149317550935817071

Offset: 1

Colin Barker, Jun 26 2015

			2080 is in the sequence because T(64) = 2080 = 66 + 78 + 91 + 105 + 120 + 136 + 153 + 171 + 190 + 210 + 231 + 253 + 276 = T(11) + ... + T(23).

Colin Barker, Table of n, a(n) for n = 1..641
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1684802,-1684802,0,0,-1,1).

Cf. A000217, A001110, A129803, A131557, A257711, A257712, A257713, A259413, A259415.

 LinearRecurrence[{1, 0, 0, 1684802, -1684802, 0, 0, -1, 1}, {2080, 414505, 28815436, 49317346, 3428789455, 698283666730, 48548229019381, 83089887991201, 5776831256176630}, 30] (* Vincenzo Librandi, Jun 27 2015 *)

Vec(-13*x*(7*x^8 +153*x^6 +31725*x^5 -9608927*x^4 +1577070*x^3 +2184687*x^2 +31725*x +160) / ((x -1)*(x^2 -36*x -1)*(x^2 +36*x -1)*(x^4 +1298*x^2 +1)) + O(x^20))

G.f.: -13*x*(7*x^8 +153*x^6 +31725*x^5 -9608927*x^4 +1577070*x^3 +2184687*x^2 +31725*x +160) / ((x -1)*(x^2 -36*x -1)*(x^2 +36*x -1)*(x^4 +1298*x^2 +1)).

A259415 Triangular numbers (A000217) that are the sum of seventeen consecutive triangular numbers.

Original entry on oeis.org

1326, 9180, 24531, 1979055, 5325216, 39529386, 106368405, 8616365901, 23185550130, 172110498456, 463127571831, 37515654714891, 100949879501796, 749369070309030, 2016457340944761, 163343152011830505, 439535752164830646, 3262752760014579156

Offset: 1

Colin Barker, Jun 26 2015

			1326 is in the sequence because T(51) = 1326 = 6 + 10 + 15 + 21 + 28 + 36 + 45 + 55 + 66 + 78 + 91 + 105 + 120 + 136 + 153 + 171 + 190 = T(3) + ... + T(19).

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,4354,-4354,0,0,-1,1).

Cf. A000217, A001110, A129803, A131557, A257711, A257712, A257713, A259413, A259414.

 LinearRecurrence[{1, 0, 0, 4354, -4354, 0, 0, -1, 1}, {1326, 9180, 24531, 1979055, 5325216, 39529386, 106368405, 8616365901, 23185550130}, 30] (* Vincenzo Librandi, Jun 27 2015 *)
Module[{nn=10^6},Select[Total/@Partition[Accumulate[Range[nn]],17,1],OddQ[ Sqrt[8#+1]]&]] (* Harvey P. Dale, Mar 19 2023 *)

Vec(-51*x*(11*x^8 +15*x^6 +154*x^5 -47593*x^4 +38324*x^3 +301*x^2 +154*x +26) / ((x -1)*(x^2 -8*x -1)*(x^2 +8*x -1)*(x^4 +66*x^2 +1)) + O(x^30))

G.f.: -51*x*(11*x^8 +15*x^6 +154*x^5 -47593*x^4 +38324*x^3 +301*x^2 +154*x +26) / ((x -1)*(x^2 -8*x -1)*(x^2 +8*x -1)*(x^4 +66*x^2 +1)).

A082840 a(n) = 4*a(n-1) - a(n-2) + 3, with a(0) = -1, a(1) = 1.

Original entry on oeis.org

-1, 1, 8, 34, 131, 493, 1844, 6886, 25703, 95929, 358016, 1336138, 4986539, 18610021, 69453548, 259204174, 967363151, 3610248433, 13473630584, 50284273906, 187663465043, 700369586269, 2613814880036, 9754889933878, 36405744855479, 135868089488041

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), Apr 14 2003

Apart from the initial -1, these are the numbers k such that the triangular number k*(k + 1)/2 is the sum of three consecutive triangular numbers - see A129803. - Brian Nowell, Nov 03 2009

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

Cf. A001571, A071954.

a:=[-1,1,8];; for n in [4..30] do a[n]:=5*a[n-1]-5*a[n-2]+a[n-3]; od; a; # G. C. Greubel, Feb 25 2019

m:=30; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( -(1-6*x+2*x^2)/((1-x)*(1-4*x+x^2)) )); // G. C. Greubel, Feb 25 2019

CoefficientList[Series[(-1+6x-2x^2)/((1-x)(1-4x+x^2)), {x, 0, 30}], x] (* Vincenzo Librandi, Apr 15 2014 *)
LinearRecurrence[{5,-5,1}, {-1,1,8}, 30] (* G. C. Greubel, Feb 25 2019 *)

is(n)=ispolygonal(3/2*n*(n+1)+4,3) || n==-1 \\ Charles R Greathouse IV, Apr 14 2014

my(x='x+O('x^30)); Vec(-(1-6*x+2*x^2)/((1-x)*(1-4*x+x^2))) \\ G. C. Greubel, Feb 25 2019

(-(1-6*x+2*x^2)/((1-x)*(1-4*x+x^2))).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 25 2019

a(n) = A001571(n) - 1. - N. J. A. Sloane, Nov 03 2009
G.f.: -(1 -6*x +2*x^2)/((1 - x)*(1 - 4*x + x^2)).
a(n) = -3/2 + (1/12)*( (a -2*b +5)*a^n + (b -2*a +5)*b^n ), with a = 2 + sqrt(3), b = 2 - sqrt(3):.
a(n) = -3/2 + (3/4)*A003500(n) - (1/4)*A003500(n-1).
a(n) = (1/2)*(A001834(n) - 3).
E.g.f.: ((1 + sqrt(3))*exp((2 + sqrt(3))*x) + (1 - sqrt(3))*exp((2 - sqrt(3))*x) - 6*exp(x))/4. - Franck Maminirina Ramaharo, Nov 12 2018

A129863 Sums of three consecutive pentagonal numbers.

Original entry on oeis.org

6, 18, 39, 69, 108, 156, 213, 279, 354, 438, 531, 633, 744, 864, 993, 1131, 1278, 1434, 1599, 1773, 1956, 2148, 2349, 2559, 2778, 3006, 3243, 3489, 3744, 4008, 4281, 4563, 4854, 5154, 5463, 5781, 6108, 6444, 6789, 7143, 7506, 7878, 8259, 8649, 9048, 9456, 9873

Offset: 0

Jonathan Vos Post, May 23 2007, May 24 2007

Arises in pentagonal number analog to A129803, Triangular numbers that are the sum of three consecutive triangular numbers. What are the pentagonal numbers which are the sum of three consecutive pentagonal numbers?

			a(0) = 6 = A000326(0) + A000326(1) + A000326(2) = 0 + 1 + 5.
a(1) = 18 = A000326(1) + A000326(2) + A000326(3) = 1 + 5 + 12.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A007667, A034961, A129803.

[(9/2)*(n^2)+(15/2)*n+6: n in [0..50]]; // Vincenzo Librandi, Aug 16 2017

Table[(3/2)*(4 + 5*n + 3*n^2), {n, 0, 100}] (* Stefan Steinerberger, May 27 2007 *)
CoefficientList[Series[3 (2 + x^2) / (1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 16 2017 *)
Total/@Partition[PolygonalNumber[5,Range[0,50]],3,1] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{3,-3,1},{6,18,39},50] (* Harvey P. Dale, Nov 22 2018 *)

a(n)=n*(9*n+15)/2+6 \\ Charles R Greathouse IV, Jun 17 2017

a(n) = P(n) + P(n+1) + P(n+2) where P(n) = A000326(n) = n*(3*n-1)/2.
a(n) = (9/2)*(n^2) + (15/2)*n + 6.
a(n) = (3*n^2 + 5*n + 4)*(3/2). - Stefan Steinerberger, May 27 2007
G.f.: 3*(2+x^2)/(1-x)^3. - Colin Barker, Feb 13 2012
From Elmo R. Oliveira, Nov 16 2024: (Start)
E.g.f.: 3*exp(x)*(3*x^2 + 8*x + 4)/2.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)

Offset corrected by Eric Rowland, Aug 15 2017

cp's OEIS Frontend

A034706 Numbers which are sums of consecutive triangular numbers.

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

A131557 Triangular numbers that are the sums of five consecutive triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A257711 Triangular numbers (A000217) that are the sum of seven consecutive triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A257712 Triangular numbers (A000217) that are the sum of eight consecutive triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A257713 Triangular numbers (A000217) that are the sum of ten consecutive triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A259413 Triangular numbers (A000217) that are the sum of eleven consecutive triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A259414 Triangular numbers (A000217) that are the sum of thirteen consecutive triangular numbers.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A259415 Triangular numbers (A000217) that are the sum of seventeen consecutive triangular numbers.

Views