A006578 -id:A006578 - cp's OEIS frontend

A332410 a(n) = 2a(n-1) - a(n-2) + a(n-5) - 2a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

0, 1, 3, 6, 11, 17, 24, 32, 41, 52, 64, 77, 91, 106, 123, 141, 160, 180, 201, 224, 248, 273, 299, 326, 355, 385, 416, 448, 481, 516, 552, 589, 627, 666, 707, 749, 792, 836, 881, 928, 976, 1025, 1075, 1126, 1179

Offset: 0

Paul Curtz, Feb 11 2020

This sequence occurs twice as a linear spoke in the hexagonal spiral constructed from A002266:
                       17  17  17  17  17  18  18
                     16  11  11  11  11  12  12  18
                   16  11   6   6   7   7   7  12  18
                 16  10   6   3   3   3   3   7  12  18
               16  10   6   3   1   1   1   4   7  12  19
             16  10   6   2   0   0   0   1   4   8  13  19
               15  10   5   2   0   0   1   4   8  13  19
                 15  10   5   2   2   2   4   8  13  19
                   15   9   5   5   5   4   8  13  19
                     15   9   9   9   9   8  13  20
                       15  14  14  14  14  14  20
a(-1-n) = 0, 1, 4, 8, 13, 19, 26, 35, 45, ... also occurs twice in the same spiral.
Difference table:
  0, 1, 3, 6, 11, 17, 24, 32, 41, 52, ... = a(n)
  1, 2, 3, 5,  6,  7,  8,  9, 11, 12, ... = A047256(n+1)
  1, 1, 2, 1,  1,  1,  1,  2,  1,  1, ... = A130782.
There is no linear spoke with three copies in this spiral. Compare with the spiral illustrated in sequence A330707 and constructed from A002265 where the same spokes occur three times: A006578, A001859 and A077043, essentially. Strictly, three times from 1, 1, 1 for A006578, from 2, 2, 2 for A001859 and from 7, 7, 7 for A077043.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,1,-2,1).

Cf. A002266, A008602, A047256, A130782, A330707.

Cf. A001859, A006578, A077043.

LinearRecurrence[{2, -1, 0, 0, 1, -2, 1}, {0, 1, 3, 6, 11, 17, 24}, 45] (* Amiram Eldar, Feb 12 2020 *)

concat(0, Vec(x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)) + O(x^50))) \\ Colin Barker, Feb 11 2020, Apr 24 2020

a(8+n) - a(8-n) = 20*n.

G.f.: x*(1 + x)*(1 + x^2 + x^3) / ((1 - x)^3*(1 + x + x^2 + x^3 + x^4)). - Colin Barker, Feb 11 2020

A104568 Triangle of numbers that are 0 or 1 mod 3.

Original entry on oeis.org

1, 3, 1, 4, 3, 1, 6, 4, 3, 1, 7, 6, 4, 3, 1, 9, 7, 6, 4, 3, 1, 10, 9, 7, 6, 4, 3, 1, 12, 10, 9, 7, 6, 4, 3, 1, 13, 12, 10, 9, 7, 6, 4, 3, 1, 15, 13, 12, 10, 9, 7, 6, 4, 3, 1, 16, 15, 13, 12, 10, 9, 7, 6, 4, 3, 1, 18, 16, 15, 13, 12, 10, 9, 7, 6, 4, 3, 1, 19, 18, 16, 15, 13, 12, 10, 9, 7, 6, 4, 3, 1

Offset: 0

Gary W. Adamson, Mar 16 2005

The matrix operations (J * R), (R * J) are commutative since J * R = R * J.
Row sums = A006578.
Rows and columns of the triangle are all 0 or 1 mod 3 terms: A032766.
A104567 row sums also = A006578.
A006578(2n-1) = A001082(2n).

			The first few rows are:
  1;
  3, 1;
  4, 3, 1;
  6, 4, 3, 1;
  7, 6, 4, 3, 1;
  9, 7, 6, 4, 3, 1;
  ...

Cf. A001082, A006578, A104566, A104567.

it:=array(1..1000): i:=1: for n from 1 to 1000 do if n mod 3 <> 2 then it[i]:=n; i:=i+1 fi: od: for j from 1 to 25 do for k from j to 1 by -1 do printf(`%d,`,it[k]) od: od: # James Sellers, Apr 09 2005

All columns (with offset); and all rows (starting from the right) are 0 or 1 mod 3 (A032766). Extract the triangle from the product J * R; J = [1; 2, 1; 1, 2, 1; 2, 1, 2, 1; ...]; R = [1; 1, 1; 1, 1, 1; ...] (infinite lower triangular matrices, with the rest zeros).

More terms from James Sellers, Apr 09 2005

A266086 Alternating sum of 9-gonal (or nonagonal) numbers.

Original entry on oeis.org

0, -1, 8, -16, 30, -45, 66, -88, 116, -145, 180, -216, 258, -301, 350, -400, 456, -513, 576, -640, 710, -781, 858, -936, 1020, -1105, 1196, -1288, 1386, -1485, 1590, -1696, 1808, -1921, 2040, -2160, 2286, -2413, 2546, -2680, 2820, -2961, 3108, -3256, 3410

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Eric Weisstein's World of Mathematics, Nonagonal Number
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A001106, A006578, A007584, A035608, A083392, A089594.

[(14*(-1)^n*n^2 + 4*(-1)^n*n - 5*(-1)^n + 5)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[((14 n^2 + 4 n - 5) (-1)^n + 5)/8, {n, 0, 44}]
CoefficientList[Series[(x - 6 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

x='x+O('x^100); concat(0, Vec(-x*(1-6*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 6*x)/((1 - x)*(1 + x)^3).
a(n) = ((14*n^2 + 4*n - 5)*(-1)^n + 5)/8.
a(n) = Sum_{k = 0..n} (-1)^k*A001106(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.

A266087 Alternating sum of 11-gonal (or hendecagonal) numbers.

Original entry on oeis.org

0, -1, 10, -20, 38, -57, 84, -112, 148, -185, 230, -276, 330, -385, 448, -512, 584, -657, 738, -820, 910, -1001, 1100, -1200, 1308, -1417, 1534, -1652, 1778, -1905, 2040, -2176, 2320, -2465, 2618, -2772, 2934, -3097, 3268, -3440, 3620, -3801, 3990, -4180

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A006578, A007586, A035608, A051682, A083392, A089594.

[(18*(-1)^n*n^2 + 4*(-1)^n*n - 7*(-1)^n + 7)/8: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[((18 n^2 + 4 n - 7) (-1)^n + 7)/8, {n, 0, 43}]
CoefficientList[Series[(x - 8 x^2)/(x^4 + 2 x^3 - 2 x - 1), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)
Accumulate[Times@@@Partition[Riffle[PolygonalNumber[11,Range[0,50]],{1,-1},{2,-1,2}],2]] (* Requires Mathematica version 10 or later *) (* or *) LinearRecurrence[{-2,0,2,1},{0,-1,10,-20},50] (* Harvey P. Dale, Aug 27 2019 *)

x='x+O('x^100); concat(0, Vec(-x*(1-8*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 8*x)/((1 - x)*(1 + x)^3).
a(n) = ((18*n^2 + 4*n - 7)*(-1)^n + 7)/8.
a(n) = Sum_{k = 0..n} (-1)^k*A051682(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.
E.g.f.: (1/4)*(9*x^2 - 11*x)*cosh(x) - (1/4)*(9*x^2 - 11*x - 7)*sinh(x). - G. C. Greubel, Jan 27 2016

A266088 Alternating sum of 12-gonal (or dodecagonal) numbers.

Original entry on oeis.org

0, -1, 11, -22, 42, -63, 93, -124, 164, -205, 255, -306, 366, -427, 497, -568, 648, -729, 819, -910, 1010, -1111, 1221, -1332, 1452, -1573, 1703, -1834, 1974, -2115, 2265, -2416, 2576, -2737, 2907, -3078, 3258, -3439, 3629, -3820, 4020, -4221, 4431, -4642

Offset: 0

Ilya Gutkovskiy, Dec 21 2015

More generally, the ordinary generating function for the alternating sum of k-gonal numbers is -x*(1 - (k - 3)*x)/((1 - x)*(1 + x)^3).

G. C. Greubel, Table of n, a(n) for n = 0..5000
OEIS Wiki, Figurate numbers
Index entries for linear recurrences with constant coefficients, signature (-2,0,2,1).

Cf. A006578, A007587, A035608, A051624, A083392, A089594.

[1+(-1)^n*(5*n^2+n-2)/2: n in [0..50]]; // Vincenzo Librandi, Dec 21 2015

Table[1 + (-1)^n (5 n^2 + n - 2)/2, {n, 0, 43}]
CoefficientList[Series[-x (1 - 9 x)/((1 - x) (1 + x)^3), {x, 0, 50}], x] (* Vincenzo Librandi, Dec 21 2015 *)

x='x+O('x^100); concat(0, Vec(-x*(1-9*x)/((1-x)*(1+x)^3))) \\ Altug Alkan, Dec 21 2015

G.f.: -x*(1 - 9*x)/((1 - x)*(1 + x)^3).
a(n) = 1 + (-1)^n*(5*n^2 + n - 2)/2.
a(n) = Sum_{k = 0..n} (-1)^k*A051624(k).
Lim_{n -> infinity} a(n + 1)/a(n) = -1.

A271937 a(n) = (7/4)n^2 + (5/2)n + (7 + (-1)^n)/8.

Original entry on oeis.org

1, 5, 13, 24, 39, 57, 79, 104, 133, 165, 201, 240, 283, 329, 379, 432, 489, 549, 613, 680, 751, 825, 903, 984, 1069, 1157, 1249, 1344, 1443, 1545, 1651, 1760, 1873, 1989, 2109, 2232, 2359, 2489, 2623, 2760, 2901, 3045, 3193, 3344, 3499, 3657, 3819, 3984, 4153

Offset: 0

Vincenzo Librandi, Apr 20 2016

Let P be a polygon with vertices (0,0), (0,2), (1,1) and (0,3/2). The number of integer points in nP is counted by this quasi-polynomial (nP is the n-fold dilation of P). See Wikipedia in Links section.
From Bob Selcoe, Sep 10 2016: (Start)
a(n) = the number of partitions in reverse lexicographic order starting with n 3's followed by n 2's; i.e., the number of partitions summing to 5n such that no part > 3 and the number of 3's digits <= the number of 2's digits.
First differences are A047346(n+1); second differences are 4 when n is even and 3 when n is odd (i.e., A010702(n+1)); third differences are 1 when n is even and -1 when n is odd. (End)

			a(1) = 5; the 5 partitions are: {3,2}; {3,1,1}; {2,2,1}; {2,1,1,1}; {1,1,1,1,1}.
a(3) = 24: floor(8/2) + floor(11/2) + floor(14/2) + floor(17/2) = 4+5+7+8 = 24.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Wikipedia, Examples of Ehrhart Quasi-Polynomials.
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,1).

First bisection (after 1) is A168235.
Second bisection is A135703 (without 0).
Cf. A006578, A047346.

[(7/4)*n^2+(5/2)*n+(7+(-1)^n)/8: n in [0..50]];

Table[(7/4) n^2 + (5/2) n + (7 + (-1)^n)/8, {n, 0, 50}]
LinearRecurrence[{2,0,-2,1},{1,5,13,24},50] (* Harvey P. Dale, Mar 23 2025 *)

Vec((1+3*x+3*x^2)/((1-x)^3*(1+x)) + O(x^99)) \\ Altug Alkan, Sep 10 2016

O.g.f.: (1 + 3*x + 3*x^2)/((1 - x)^3*(1 + x)).
E.g.f.: (7 + 34*x + 14*x^2)*exp(x)/8 + exp(-x)/8.
a(n) = 2*a(n-1) - 2*a(n-3) + a(n-4).
a(2*k) = k*(7*k + 5) + 1, a(2*k+1) = (k + 1)*(7*k + 5).
From Bob Selcoe, Sep 10 2016 (Start):
a(n) = (n+1)^2 + A006578(n).
a(n) = a(n-1) + A047346(n+1).
a(n) = Sum_{j=0..n} floor((2n+3j+2)/2).
(End)

Edited and extended by Bruno Berselli, Apr 20 2016

cp's OEIS Frontend

A332410 a(n) = 2*a(n-1) - a(n-2) + a(n-5) - 2*a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

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A104568 Triangle of numbers that are 0 or 1 mod 3.

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A266086 Alternating sum of 9-gonal (or nonagonal) numbers.

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Magma

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A266087 Alternating sum of 11-gonal (or hendecagonal) numbers.

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Magma

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A266088 Alternating sum of 12-gonal (or dodecagonal) numbers.

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Magma

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A271937 a(n) = (7/4)*n^2 + (5/2)*n + (7 + (-1)^n)/8.

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A332410 a(n) = 2a(n-1) - a(n-2) + a(n-5) - 2a(n-6) + a(n-7) with a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=11, a(5)=17, a(6)=24.

A271937 a(n) = (7/4)n^2 + (5/2)n + (7 + (-1)^n)/8.