A034706 -id:A034706 - cp's OEIS frontend

A034705 Numbers that are sums of consecutive squares.

0, 1, 4, 5, 9, 13, 14, 16, 25, 29, 30, 36, 41, 49, 50, 54, 55, 61, 64, 77, 81, 85, 86, 90, 91, 100, 110, 113, 121, 126, 135, 139, 140, 144, 145, 149, 169, 174, 181, 190, 194, 196, 199, 203, 204, 221, 225, 230, 245, 255, 256, 265, 271, 280, 284, 285, 289, 294, 302

Offset: 1

Erich Friedman

nonn

Also, differences of any pair of square pyramidal numbers (A000330). These could be called "truncated square pyramidal numbers". - Franklin T. Adams-Watters, Nov 29 2006

If n is the sum of d consecutive squares up to m^2, n = A000330(m) - A000330(m-d) = d*(m^2 - (d-1)*m + (d-1)*(2*d-1)/6) <=> m^2 - (d-1)*m = c := n/d - (d-1)*(2*d-1)/6 <=> m = (d-1)/2 + sqrt((d-1)^2/4 + c) which must be an integer. Moreover, A000330(x) >= x^3/3, so m and d can't be larger than (3*n)^(1/3). - M. F. Hasler, Jan 02 2024

			All squares (A000290: 0, 1, 4, 9, ...) are in this sequence, since "consecutive" in the definition means a subsequence without interruption, so a single term qualifies.
5 = 1^2 + 2^2 = A000330(2) is in this sequence, and similarly 13 = 2^2 + p3^2  = A000330(3) - A000330(1) and 14 = 1^2 + 2^2 + 3^2 = A000330(3), etc.

Cf. A000290, A000330, A034706.
Cf. A217843-A217850 (sums of consecutive powers 3 to 10).
Cf. A368570 (first of each pair of consecutive integers in this sequence).

import Data.Set (deleteFindMin, union, fromList); import Data.List (inits)
a034705 n = a034705_list !! (n-1)
a034705_list = f 0 (tail $ inits $ a000290_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

nMax = 1000; t = {0}; Do[k = n; s = 0; While[s = s + k^2; s <= nMax, AppendTo[t, s]; k++], {n, Sqrt[nMax]}]; t = Union[t] (* T. D. Noe, Oct 23 2012 *)

{is_A034705(n)= for(d=1,sqrtnint(n*3,3), my(b = (d-1)/2, s = n/d - (d-1)*(d*2-1)/6 + b^2); denominator(s)==denominator(b)^2 && issquare(s, &s) && return(b+s)); !n} \\ Return the index of the largest square of the sum (or 1 for n = 0) if n is in the sequence, else 0. - M. F. Hasler, Jan 02 2024

import heapq
from itertools import islice
def agen(): # generator of terms
    m = 0; h = [(m, 0, 0)]; nextcount = 1; v1 = None
    while True:
        (v, s, l) = heapq.heappop(h)
        if v != v1: yield v; v1 = v
        if v >= m:
            m += nextcount*nextcount
            heapq.heappush(h, (m, 1, nextcount))
            nextcount += 1
        v -= s*s; s += 1; l += 1; v += l*l
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 60))) # Michael S. Branicky, Jan 06 2024

Terms a(1..10^4) double-checked with independent code by M. F. Hasler, Jan 02 2024

A322468 Numbers that are sums of consecutive tetrahedral numbers.

Original entry on oeis.org

0, 1, 4, 5, 10, 14, 15, 20, 30, 34, 35, 55, 56, 65, 69, 70, 84, 91, 111, 120, 121, 125, 126, 140, 165, 175, 195, 204, 205, 209, 210, 220, 260, 285, 286, 295, 315, 325, 329, 330, 364, 369, 385, 425, 455, 460, 480, 490, 494, 495, 505, 506, 560, 589, 645, 650, 671, 680, 700

Offset: 1

Ilya Gutkovskiy, Dec 09 2018

nonn

			209 = sum_{k=2..7} A000292(k) so 209 is in the list. 295=sum_{k=5..8} A000292(k), so 295 is in the list.

R. J. Mathar, Table of n, a(n) for n = 1..603
Eric Weisstein's World of Mathematics, Tetrahedral Number

Cf. A000292 (tetrahedral numbers, a subsequence), A000330 (subsequence), A006003 (subsequence), A005894 (subsequence).

Other sums of consecutive numbers: A034705 (squares), A034706 (triangular numbers), A322479 (square pyramidal numbers), A322610 (centered triangular numbers), A322611 (centered square numbers).

tet[n_] := n (n + 1) (n + 2)/6; nMax = 700; t = {0}; Do[k = n; s = 0; While[s = s + tet[k]; s <= nMax, AppendTo[t, s]; k++], {n, (6*nMax)^(1/3) + 1}]; t = Union[t] (* Amiram Eldar, Dec 09 2018 after T. D. Noe at A034705 *)
anmax = 1000; nmax = Floor[(6*anmax)^(1/3)] + 1; Select[Union[Flatten[Table[Sum[k*(k + 1)*(k + 2)/6, {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)

A322610 Numbers that are sums (of a nonempty sequence) of consecutive centered triangular numbers.

Original entry on oeis.org

1, 4, 5, 10, 14, 15, 19, 29, 31, 33, 34, 46, 50, 60, 64, 65, 77, 85, 96, 106, 109, 110, 111, 136, 141, 149, 160, 166, 170, 174, 175, 194, 195, 199, 226, 235, 245, 255, 258, 259, 260, 274, 302, 304, 316, 330, 335, 354, 361, 364, 365, 368, 369, 394, 409, 411, 434, 440, 460, 471, 490, 496, 500

Offset: 1

Ilya Gutkovskiy, Dec 20 2018

nonn

Eric Weisstein's World of Mathematics, Centered Triangular Number

Cf. A005448, A005918 (2 consec), A006003, A034706, A320728, A322468, A322611, A322638, A322640.

anmax = 1000; nmax = Floor[Sqrt[2*anmax/3]] + 1; Select[Union[Flatten[Table[Sum[(3*k^2 + 3*k + 2)/2, {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)

A322636 Numbers that are sums of consecutive heptagonal numbers (A000566).

Original entry on oeis.org

0, 1, 7, 8, 18, 25, 26, 34, 52, 55, 59, 60, 81, 89, 107, 112, 114, 115, 136, 148, 170, 188, 189, 193, 195, 196, 235, 248, 260, 282, 286, 300, 307, 308, 337, 341, 342, 396, 403, 424, 430, 448, 449, 455, 456, 469, 521, 530, 540, 572, 585, 616, 619, 628, 637, 644, 645, 684, 697

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Heptagonal Number

Cf. A000566, A002413, A034705, A034706, A319184, A319185, A322637.

N:= 1000: # for terms up to N
Hepta:= [seq(n*(5*n-3)/2,n=0..floor((3+sqrt(9+40*N))/10))]:
PS:= ListTools:-PartialSums(Hepta):
S:= select(`<=`,{0,seq(seq(PS[i]-PS[j],j=1..i-1),i=1..nops(PS))},N):
sort(convert(S,list)); # Robert Israel, May 22 2025

terms = 59;
nmax = 17; kmax = 9; (* empirical *)
T = Table[n(5n-3)/2, {n, 0, nmax}];
Union[T, Table[k MovingAverage[T, k], {k, 2, kmax}]//Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

A322637 Numbers that are sums of consecutive octagonal numbers (A000567).

Original entry on oeis.org

0, 1, 8, 9, 21, 29, 30, 40, 61, 65, 69, 70, 96, 105, 126, 133, 134, 135, 161, 176, 201, 222, 225, 229, 230, 231, 280, 294, 309, 334, 341, 355, 363, 364, 401, 405, 408, 470, 481, 505, 510, 531, 534, 539, 540, 560, 621, 630, 645, 681, 695, 735, 736, 749, 756, 764, 765, 814, 833, 846

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Octagonal Number

Cf. A000567, A002414, A034705, A034706, A319184, A319185, A322636.

N:= 1000: # for terms up to N
Octa:= [seq(n*(3*n-2),n=0..floor((1+sqrt(1+3*N))/3))]:
PS:= ListTools:-PartialSums(Octa):
S:= select(`<=`,{0,seq(seq(PS[i]-PS[j],j=1..i-1),i=1..nops(PS))},N):
sort(convert(S,list)); # Robert Israel, May 22 2025

terms = 60;
nmax = 17; kmax = 9; (* empirical *)
T = Table[n(3n-2), {n, 0, nmax}];
Union[T, Table[k MovingAverage[T, k], {k, 2, kmax}]//Flatten][[1 ;; terms]] (* Jean-François Alcover, Dec 26 2018 *)

A334007 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

Original entry on oeis.org

1, 10, 2180, 10053736, 13291443468940

Offset: 1

Ilya Gutkovskiy, Apr 12 2020

			Let S(k, m) denote the sum of m triangular numbers starting from k(k+1)/2. We have
a(1) = S(1, 1);
a(2) = S(4, 1) = S(1, 3);
a(3) = S(31, 4) = S(27, 5) = S(9, 15);
a(4) = S(945, 22) = S(571, 56) = S(968, 21) = S(131, 266);
a(5) = S(4109, 38947) = S(25213, 20540) = S(10296, 32943) = S(32801, 15834) = S(31654, 16472).

Eric Weisstein's World of Mathematics, Triangular Number
Index to sequences related to polygonal numbers

Cf. A000217, A054859, A068314, A034706, A050943, A186337, A298467, A307666, A309783, A334008, A334010, A334011, A334012.

a(5) from Giovanni Resta, Apr 13 2020

A319184 Numbers that are sums of consecutive pentagonal numbers.

Original entry on oeis.org

0, 1, 5, 6, 12, 17, 18, 22, 34, 35, 39, 40, 51, 57, 69, 70, 74, 75, 86, 92, 108, 117, 120, 121, 125, 126, 145, 156, 162, 176, 178, 190, 195, 196, 209, 210, 213, 247, 248, 262, 270, 279, 282, 287, 288, 321, 330, 354, 365, 376, 386, 387, 399, 404, 405, 424, 425, 438, 457, 475

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Pentagonal Number

Cf. A000326, A002411, A034705, A034706, A319185, A322636, A322637.

anmax = 1000; nmax = Floor[Sqrt[2*anmax/3]] + 1; Select[Union[Flatten[Table[Sum[k*(3*k-1)/2, {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)
Module[{nn=20,pn},pn=PolygonalNumber[5,Range[0,nn]];Take[Union[Flatten[Table[Total/@Partition[pn,d,1],{d,nn}]]],60]] (* Harvey P. Dale, Jun 22 2025 *)

A307614 Number of partitions of the n-th triangular number into consecutive positive triangular numbers.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1

Offset: 1

Ilya Gutkovskiy, Apr 18 2019

nonn

			A000217(4) = 10 = 1 + 3 + 6, so a(4) = 2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000217, A034706, A072964, A196010, A288126, A298858, A307608.

a(n) = [x^(n*(n+1)/2)] Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k*(k+1)/2).

A307666 Number of partitions of n into consecutive positive triangular numbers.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Ilya Gutkovskiy, Apr 20 2019

nonn

Equivalently, number of ways n can be expressed as the difference between two tetrahedral numbers. - Charlie Neder, Apr 24 2019

Records: a(10)=2, a(2180)=3, a(10053736)=4. - Robert Israel, Aug 20 2019

			10 = 1 + 3 + 6, so a(10) = 2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000217, A001227, A007294, A024940, A034706, A296338, A307614, A309783.

N:= 100:
V:= Vector(N):
for i from 1 while i*(i+1)/2 <= N do
  s:= i*(i+1)*(i+2)/6;
  for j from i-1 to 0 by -1 do
    t:= j*(j+1)*(j+2)/6;
    if s-t > N then break fi;
    V[s-t]:= V[s-t]+1
  od;
od:
convert(V,list); # Robert Israel, Aug 20 2019

G.f.: Sum_{i>=1} Sum_{j>=i} Product_{k=i..j} x^(k*(k+1)/2).

A319185 Numbers that are sums of consecutive hexagonal numbers (A000384).

Original entry on oeis.org

0, 1, 6, 7, 15, 21, 22, 28, 43, 45, 49, 50, 66, 73, 88, 91, 94, 95, 111, 120, 139, 153, 154, 157, 160, 161, 190, 202, 211, 230, 231, 245, 251, 252, 273, 276, 277, 322, 325, 343, 350, 364, 365, 371, 372, 378, 421, 430, 435, 463, 475, 496, 503, 507, 518, 524, 525, 554, 561

Offset: 1

Ilya Gutkovskiy, Dec 21 2018

nonn

Eric Weisstein's World of Mathematics, Hexagonal Number

Cf. A000384, A002412, A034705, A034706, A319184, A322636, A322637.

anmax = 1000; nmax = Floor[Sqrt[anmax/2]] + 1; Select[Union[Flatten[Table[Sum[k*(2*k-1), {k, i, j}], {i, 0, nmax}, {j, i, nmax}]]], # <= anmax &] (* Vaclav Kotesovec, Dec 21 2018 *)

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A034705 Numbers that are sums of consecutive squares.

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A322468 Numbers that are sums of consecutive tetrahedral numbers.

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A322610 Numbers that are sums (of a nonempty sequence) of consecutive centered triangular numbers.

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A322636 Numbers that are sums of consecutive heptagonal numbers (A000566).

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A322637 Numbers that are sums of consecutive octagonal numbers (A000567).

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A334007 a(n) is the least integer that can be expressed as the sum of one or more consecutive nonzero triangular numbers in exactly n ways.

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A319184 Numbers that are sums of consecutive pentagonal numbers.

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A307614 Number of partitions of the n-th triangular number into consecutive positive triangular numbers.

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A307666 Number of partitions of n into consecutive positive triangular numbers.

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